Academic literature on the topic 'Moderate Rayleigh numbers'

The natural convection at low and moderate Rayleigh numbers (Ra) incylindrical horizontal annuli with imposed temperatures in both surfaces isnumerically studied. This flow inside concentric cylinders classic configuration has a wide range of practical and technological applications, which justifies its growing studies efforts. In this work, the governing equations are discretized by the volume finite technique over a staggered grid, with second-order accuracy in space and time. The flow pattern is presented by several Rayleigh numbers, with an analysis of the heat transfer coefficient and flow properties. Furthermore, a three-dimensional field is shown at a moderate Ra number. The results showed a good agreement with the experimental data.

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

2007/2008
In this thesis we present the results of an extensive campaign of direct numerical simulations of Rayleigh-B\'enard convection at high Prandtl numbers ($10^{-1}\leq Pr \leq 10^4$) and moderate Rayleigh numbers ($10^{5}\leq Pr \leq 10^9$). The computational domain is a cylindrical cell of aspect-ratio (diameter over cell height) $\Gamma=1/2$, with the no-slip condition imposed to the boundaries. By scaling the results, we find a $1/\sqrt{Pr}$ correction to apply to the free-fall velocity, obtaining a more appropriate representation of the large scale velocity at high $Pr$. We investigate the Nusselt and the Reynolds number dependence on $Ra$ and $Pr$, comparing the results to previous numerical and experimental work. At high $Pr$ the scaling behavior of the Nusselt number with respect to $Ra$ is generally consistent with the power-law exponent $0.309$. The Nusselt number is independent of $Pr$, even at the highest $Ra$ simulated. The Reynolds number scales as $Re\sim \sqrt{Ra}/Pr$, neglecting logarithmic corrections. We analyze the global and local features of viscous and thermal boundary layers and their scaling behavior with respect to Rayleigh and Prandtl numbers, and with respect to Reynolds and Peclet numbers. We find that the flow approaches a saturation regime when Reynolds number decreases below the critical value $Re_s\simeq 40$. The thermal boundary layer thickness turns out to increase slightly even when the Peclet number increases. We explain this behavior as a combined effect of the Peclet number and the viscous boundary layer influences. The range of $Ra$ and $Pr$ simulated contains steady, periodic and turbulent solutions. A rough estimate of the transition from steady to unsteady flow is obtained by monitoring the time-evolution of the system until it reaches stationary solutions ($Ra_U\simeq 7.5 \times 10^6$ at $Pr=10^3$). We find multiple solutions as long-term phenomena at $Ra=10^8$ and $Pr=10^3$ which, however, do not result in significantly different Nusselt number. One of these multiple solutions, even if stable for a long time interval, shows a break in the mid-plane symmetry of the temperature profile. The result is similar to that of some non-Boussinesq effects. We analyze the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. We also describe how the behavior of the flow structures changes for increasing $Pr$. A wide variety of large-scale circulations and plumes structures are found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other solutions, at lower $Pr$, the mean flow generally consists of two opposite toroidal structures; at higher $Pr$, the flow is organized in multi-cell structures extending mostly in the vertical direction. At high $Pr$, plumes detach from sheet-like structures. The different large-scale-structure signatures are generally reflected in the data trends with respect to $Ra$, but not in those with respect to $Pr$. In particular, the Nusselt number is independent of $Pr$, even when the flow structures appear strongly different varying $Pr$. In order to assess the reliability of the data-set we perform a systematic analysis of the error affecting the data. Refinement grid analysis is extensively applied.
---------------------------------------------------------------------------------------- In questa tesi presentiamo i risultati di un'estensiva campagna di simulazioni numeriche dirette della convezione di Rayleigh-B\'enard ad alti numeri di Prandtl ($10^{-1}\leq Pr \leq 10^4$) e moderati numeri di Rayleigh ($10^{5}\leq Pr \leq 10^9$). Il dominio computazionale \`e una cella cilindrica di allungamento (diametro su altezza cella) $\Gamma=1/2$, con condizioni di non-slittamento ai contorni. Scalando i risultati, troviamo una correzione di $1/\sqrt{Pr}$ da applicare alla velocit\`a di caduta libera, ottenendo una rappresentazione pi\`u appropriata della velocit\`a di larga scala ad elevati $Pr$. Investighiamo la dipendenza del numero di Nusselt e del numero di Reynolds da $Ra$ e $Pr$, comparando i risultati con precedenti lavori numerici e sperimentali. Ad elevati $Pr$ il comportamento di scala del numero di Nusselt rispetto a $Ra$ \`e generalmente compatibile con l'esponente di legge di potenza $0.309$. Il numero di Nusselt \`e indipendente da $Pr$, anche per il pi\`u alto $Ra$ simulato. Il numero di Reynolds scala come $Re\sim \sqrt{Ra}/Pr$, a meno di correzioni logaritmiche. Analizziamo le caratteristiche locali e globali degli strati limite viscosi e termici, ed il loro comportamento di scala rispetto ai numeri Rayleigh e Prandtl, e rispetto ai numeri Reynolds e Peclet. Troviamo che il flusso approccia un regime di saturazione quando il numero di Reynolds scende sotto il valore critico $Re_s\simeq 40$. Lo spessore dello strato limite termico comincia a crescere leggermente anche quando in numero di Peclet aumenta. Spieghiamo questo comportamento come un effetto combinato delle influenze del numero di Peclet e dello strato limite viscoso. L'intervallo di $Ra$ e $Pr$ simulato contiene soluzioni stazionarie, periodiche e turbolente. Una stima approssimata della transizione da flusso stazionario a non stazionario \`e ottenuta monitorando l'evoluzione temporale del sistema fino al raggiungimento di soluzioni stazionarie o statisticamente stazionarie ($Ra_U\simeq 7.5 \times 10^6$ a $Pr=10^3$). Troviamo soluzioni multiple come fenomeni di lungo termine a $Ra=10^8$ e $Pr=10^3$ che, comunque, non comportano differenze significative nel numero di Nusselt. Una di queste soluzioni multiple, anche se stabile per un lungo intervallo di tempo, mostra una rottura della simmetria del profilo di temperatura rispetto al piano mediano. Il risultato \`e simile a quello di alcuni effetti di non-Boussinesq. Analizziamo le strutture del flusso nelle fasi di transizione tramite visualizzazioni dirette dei campi di velocit\`a e temperatura. Descriviamo inoltre come il comportamento delle strutture del flusso cambia al crescere di $Pr$. Un'ampia variet\`a di circolazioni di larga scala e strutture a pennacchio vengono trovate. La circolazione a singolo anello \`e caratteristica solo delle soluzioni stazionarie e periodiche. Per le altre soluzioni, a $Pr$ pi\`u bassi, il flusso medio \`e generalmente composto da due strutture toroidali opposte; a $Pr$ pi\`u alti, il flusso \`e organizzato in strutture multi-cellulari che si estendono maggiormente in direzione verticale. Ad alti $Pr$, pennacchi si staccano da strutture simili a fogli. Le impronte delle differenti strutture di larga scala si riflettono generalmente nell'andamento dei dati rispetto a $Ra$, ma non rispetto a $Pr$. In particolare, il numero di Nusselt \`e indipendente da $Pr$, anche quando le strutture del flusso appaiono molto differenti al variare di $Pr$. Per stabilire l'affidabilit\`a dell'insieme dei dati, effettuiamo un'analisi sistematica degli errori a cui i dati sono soggetti. L'analisi di raffinamento della griglia \`e largamente applicata.
XXI Ciclo
1976

The present chapter investigates both the effects of moving loads and of stochastic wind on the steady-state vibration of a first mode Rayleigh elastic beam. The beam is assumed to lay on foundations (bearings) that are characterized by fractional-order viscoelastic material. The viscoelastic property of the foundation is modeled using the constitutive equation of Kelvin-Voigt type, which contain fractional derivatives of real order. Based to the stochastic averaging method, an analytical explanation on the effects of the viscoelastic physical properties and number of the bearings, additive and parametric wind turbulence on the beam oscillations is provided. In particular, it is found that as the number of bearings increase, the resonant amplitude of the beam decreases and shifts towards larger frequency values. The results also indicate that as the order of the fractional derivative increases, the amplitude response decreases. We are also demonstrated that a moderate increase of the additive and parametric wind turbulence contributes to decrease the chance for the beam to reach the resonance. The remarkable agreement between the analytical and numerical results is also presented in this chapter.

A numerical investigation has been conducted on a heated circular disc immersed in quiescent air in an enclosed environment and for rarefied conditions. For moderate to low Rayleigh numbers, laminar natural convection can be described by two regimes and a transitional region which separates the diffusive limit and laminar boundary layer, or convective limit. A practical scenario was simulated which involved lowering Rayleigh number and increasing Knudsen number of the disc simultaneously to examine the convective-diffusive relationship under a rarefied condition. The numerical model used to solve the continuum equations has been extended using a first order slip condition and assessed based on the available experimental data from literature. Although similar convective-diffusive regimes exist, there is a divergence from the classical continuum diffusive limit when operating in rarefied conditions with Knudsen numbers of order 10−3 or higher. The conditions for this second transitional region in heat transport from the surface have been highlighted and have been attributed to the temperature jump condition. This has resulted in recommendations on the limitation of existing analytical heat transfer models when applied to rarefied conditions. The findings also have practical significance to thermal metrology and aerospace industries where buoyancy induced flows and low ambient pressures are frequently experienced.

Natural convective heat transfer rate from an isothermal flat plate inclined at moderate angles to the vertical has been numerically studied. When the plate is wide compared to its height the flow can be adequately modeled by assuming two-dimensional flow. However, when the width of the plate is relatively small compared to its height, the heat transfer rate can be considerably greater than that predicted by these two-dimensional flow results. The heat transfer from a narrow isothermal plate embedded in a plane adiabatic surface, the adiabatic surface being in the same plane as the heated plate and inclined at an angle to the vertical has been numerically considered. Results for both positive and negative inclination angles have been numerically determined here. Attention was restricted to results for a Prandtl number of 0.7; this being approximately the value existing in the application that originally motivated this study. It has been assumed that the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces, this having been treated by using the Boussinesq approach. It has also been assumed that the flow is symmetrical about the vertical centre-plane of the plate. The solution has been obtained by numerically solving the full three-dimensional form of the governing equations, these equations being written in dimensionless form. The solution was obtained using a commercial finite element method based code, FIDAP. The solution has the Rayleigh number, the dimensionless plate width, the angle of inclination, and the Prandtl number as parameters. Results have been obtained for Rayleigh numbers between 103 and 107 for ratios of the plate width to the plate height of between 0.3 and 1.5 and for angles of inclination between +45° and −45°.

Internal natural convective heat transfer from a thin walled, vertical cylinder with an exposed vertical surface is investigated numerically. The top and bottom end faces are assumed isothermal. This setup approximates the vertical wellbore of a christmas tree for simulating cool-down of subsea oil and gas equipment during shutdown operations. The primary objective of this study is to determine the cooling rate of the interior fluid and the onset of fluid rotation caused by the two non-adiabatic surfaces as a function of Biot (Bi) at the vertical cylindrical wall. The flow is assumed to be three-dimensional, non-steady, and transitional with constant fluid properties except for the density variation with temperature. This latter effect gives rise to the buoyancy forces; being treated by using the Boussinesq approach. The solution is obtained by numerically solving the governing equations; these equations were written in terms of dimensionless variables. The solution is obtained using a commercial finite element method based-code, COMSOL Multiphysics. The specific application that motivated this investigation involved a range of Prandtl numbers (Pr) from 0.7, to 168. The results for these cases are presented herein. In addition, a range of other governing parameters has been considered. From the numerical results for small and moderate values of Rayleigh number, it is observed that steady state solutions (i.e. temperature and velocity) are stratified at the source temperature region regardless of the Biot number.

Heat transfer induced by buoyancy from a pipe buried in a semi-infinite porous medium with a superimposed fluid layer has been numerically examined in this study. Due to the complexity involved, finite difference method along with body-fitted coordinate systems has been employed. The Brinkman-extended Darcy equations are used to model flow in the porous medium while Navier-Stokes equations are used for the fluid layer. The conditions applied at the interface between the fluid and porous layers are the continuity of temperature, heat flux, normal and tangential velocity, shear stress and pressure. A parametric study has been performed to investigate the effects of Rayleigh number, Prandtl number, Darcy number, and fluid layer thickness on the flow patterns and heat transfer rates. The results show that heat transfer increases with the Rayleigh number, but the convective strength decreases with the Darcy number. The heat transfer rate is smaller when the superimposed fluid is air instead of water. For a porous layer with Da ≤ 0.0005 and an overlaying fluid layer thickness of L/ri ≥ 1, convection is initiated in the fluid layer and it may develop into multiple recirculating cells at a moderate Rayleigh number (i.e., Ra ≤ 104), and may further develop into a single cell at a higher Rayleigh number of 105.

Higher turbine inlet temperatures are a common method of increasing the thermal efficiency of modern gas turbines. This development not only generates the need for more efficient turbine blade cooling but also demands a more profound knowledge of the mechanically and thermally stressed parts of the rotor. In order to determine thermal stresses from the temperature distribution in the rotor of a gas turbine, one has to encounter the convective heat transfer in rotor cavities. In the special case of a completely closed gas-filled rotating annulus the convective flow is governed by strong natural convection. As shown in a previous paper by the authors, and for example by Owen, the presence of turbulence and its inclusion in the modeling of the flow has been found to cause significant differences in the flow development in rotating annuli. This influence in the special case of a closed rotating annulus has been recently investigated by the authors for a moderately high Rayleigh-Number. Based on this work an investigation was undertaken focusing on the development of turbulence and turbulence related changes in the flow structure for increasing Rayleigh-Numbers. The flow is investigated numerically using a three-dimensional Navier-Stokes solver, based on a pressure correction scheme. To account for the turbulence, a low-Reynolds-number k-ε-model is employed. This model is complemented by an additional term for turbulence production due to buoyancy. The results are compared with experiments performed at the Institute of Steam and Gas Turbines. The computations demonstrate the considerable influence on the overall heat transfer as well as on the local heat transfer distribution.

The historical data for circular jets indicates that the incipient cavitation number increases with the diameter of the jet. This trend is not explained by the classic cavitation theory which expects incipient cavitation number to remain constant regardless of the jet diameter, flow parameters, or water quality. This paper explores the origins of cavitation scale effects and explains the correlation between the incipient cavitation number, jet diameter, and nuclei size. This is accomplished through turbulence-resolving CFD simulations of the jet flow field at three length scales and Rayleigh-Plesset bubble dynamics for three nuclei sizes. The numerical simulations show that incipient cavitation number (σi) changes significantly as the size of the jet is altered while the Reynolds number and the value of the minimum pressure coefficient are held constant. Larger nuclei bubbles (100μm) exhibit an increase in σi with jet diameter, while moderate (50μm) and small (10μm) nuclei bubble exhibit a decrease in σi as jet diameter increases. The value of σi associated with a small jet was similar for all nuclei sizes. As the jet increased in size, the disparity between the values of σi associated with each nuclei size was found to increase substantially. The equilibrium form of the Rayleigh-Plesset equation was used to derive a correction to the classic theory of cavitation inception. This correction is a function of initial nuclei size and the dynamic head of the flow. As either the nuclei properties or dynamic head of the fluid change, the magnitude of the correction term will also change. This correction to the classic cavitation theory was used to make predictions of how σi will change as length scale and Reynolds number are altered. These equilibrium predictions were found to be in good agreement with the numerical simulations of cavitation inception for large and moderate (100μm and 50μm) nuclei bubbles. Comparisons with the small (10μm) nuclei bubbles indicate that the inertial terms are quite significant for these bubbles, resulting in large discrepancies between the full numerical solution and the equilibrium predictions. In general, the equilibrium scaling relations show that as the length scale of a flow is held constant and the Reynolds number is increased, σi will converge to −CPmin. The scaling relations also show that when Reynolds number is held constant and the length scale of a flow is increased, σi will depart from −CPmin.

Abstract Flow in a rectangular enclosure with a square horizontal cross-section and with a uniform heat flux applied across the lower horizontal surface and with the upper horizontal surface cooled to a uniform low temperature has been numerically studied. The vertical side-walls of the enclosure are adiabatic. It has been assumed that the flow is laminar and that the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces. The unsteady form of the governing equations, written in terms of the vector potential and vorticity vector functions and expressed in dimensionless form, have been solved using a finite-difference procedure based on the equations. The solution was started with no flow in the enclosure. The solution, in general, has the following parameters: the heat flux Rayleigh number Raq, the Prandtl number and the size A of the square cross-sectional shape compared to the height of the enclosure. Results have only been obtained for a Prandtl number of 0.7. Results for values of A between 0.5 and 3 for various relatively low and moderate values of Rayleigh number (up to 40000) have been obtained. The results have been used to determine the effect of A on the value of Raq below which there is no fluid motion in the enclosure and to examine the various flow patterns that arise as the value of Raq is increased with various values of A. The effect of A on the variation of the mean dimensionless lower surface temperature with Ra has also been examined.

In this work numerical experiment was performed for studying the heat transfer and thermodynamic performance of the melting process in a bottom heated square cavity. The bottom wall is maintained at a temperature higher that the melting temperature of the PCM, while all other walls are perfectly insulated. The transient numerical simulations were performed for melting Gallium, (a low Prandtl number PCM with high latent heat to density ratio) at moderate Rayleigh number (Ra ≊ 105). The transient numerical model consist of solving coupled continuity, momentum and energy equation in the unstructured formulation using the PISO algorithm. In this work, the fixed grid, source-based enthalpy-porosity approach has been adopted. The heat transfer performance of the melting process was analyzed by studying the evolution of global fluid fraction, Nusselt number at the hot wall, volume averaged normalised flow kinetic energy with time. The thermodynamic performance is analyzed by calculating local entropy generation rates considering both irreversibility due to finite temperature gradient and viscous dissipation. The values of second law efficiency clearly shows that the current thermal design of the phase-change heat accumulators are very close to the ideal design.